Solveeit Logo
Login

Question

Mathematics Question on Differential equations

Let the population of rabbits surviving at a time tt be governed by the differential equation dp(t)dt=12p(t)−200.\frac {dp(t)}{dt}=\frac {1}{2} p(t)-200. If p(0)=100,p(0)=100, then p(t)p(t) is equal to

A

400−300 et/2400-300\,e^{t/2}

B

300−200 e−t/2300-200\,e^{-t/2}

C

600−500 et/2600-500\,e^t/2

D

400−300 et/2400-300\,e^t/2

Answer

400−300 et/2400-300\,e^{t/2}

Explanation

Solution

dp(t)dt=12p(t)−200\frac{d p(t)}{d t}=\frac{1}{2} p(t)-200
∫d(p(t))(12p(t)−200)=∫dt\int \frac{d(p(t))}{\left(\frac{1}{2} p(t)-200\right)}=\int dt
2 log⁡(p(t)2−200)=t+c2 \,\log \left(\frac{p(t)}{2}-200\right)=t+c
p(t)2−200=et2k\frac{p(t)}{2}-200=e^{\frac{t}{2}} k
Using given condition p(t)=400−300 et/2p(t)=400-300 \,e^{t / 2}